Universal Quantum (Semi)groups and Hopf Envelopes

Abstract

We prove that, in case A(c) = the FRT construction of a braided vector space (V,c) admits a weakly Frobenius algebra \(\mathfrak {B}\) (e.g. if the braiding is rigid and its Nichols algebra is finite dimensional), then the Hopf envelope of A(c) is simply the localization of A(c) by a single element called the quantum determinant associated with the weakly Frobenius algebra. This generalizes a result of the author together with Gastón A. García in Farinati and García (J. Noncommutative Geom. 14(3), 879–911, 2020), where the same statement was proved, but with extra hypotheses that we now know were unnecessary. Along the way, we describe a concrete construction for a universal bialgebra associated to a finite dimensional vector space together V with some algebraic structure given by a family of maps \(\{f_{i}:V^{\otimes n_{i}} \to V^{\otimes m_{i}}\}_{i\in I}\). The Dubois-Violette and Launer Hopf algebra and the co-quasi triangular property of the FRT construction play a fundamental role in the proof.